Rigorous computations for infinite dimensional dynamical systems - - PowerPoint PPT Presentation

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Rigorous computations for infinite dimensional dynamical systems

Southern Ontario Dynamics Day Toronto April 12th, 2013

Jean-Philippe Lessard

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What is a dynamical system?

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What is a dynamical system?

Popular answer: a math subject that produces beautiful pictures!

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What is a dynamical system?

Popular answer: a math subject that produces beautiful pictures!

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What is a dynamical system?

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What is a dynamical system?

Informal answer: a system that evolves with time

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What is a dynamical system?

Informal answer: a system that evolves with time

The dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space.

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What is a dynamical system?

Informal answer: a system that evolves with time

The dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space.

finance weather prediction fluids material science population dynamics

chemical reactions

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What is a dynamical system?

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What is a dynamical system?

Formal answer: a math definition

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(T, M, Φ)

A dynamical system is a tuple

T : monoid (time) M : set (state space) Φ : map (evolution function)

Φ : T × M → M

satisfying the two following properties

⇥ ⇧ ⇧ ⌅ ⇧ ⇧ ⇤ Φ(t2, Φ(t1, x)) = Φ(t1 + t2, x)

Φ(0, x) = x

∀ x ∈ M and ∀ t1, t2 ∈ T

What is a dynamical system?

Formal answer: a math definition

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(T, M, Φ)

A dynamical system is a tuple

T : monoid (time) M : set (state space) Φ : map (evolution function)

Φ : T × M → M

satisfying the two following properties

⇥ ⇧ ⇧ ⌅ ⇧ ⇧ ⇤ Φ(t2, Φ(t1, x)) = Φ(t1 + t2, x)

Φ(0, x) = x

∀ x ∈ M and ∀ t1, t2 ∈ T

What is a dynamical system?

Formal answer: a math definition

In case the state space M is a function space, we have an infinite dimensional dynamical system !

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Examples

• 1. Finite dimensional discrete dynamical systems

f(x) = ⇢ 2x, for x ∈ [0, 1

2)

2(1 − x), for x ∈ [ 1

2, 1]

Φ : T × M → M (n, x) 7! Φ(n, x) = f n(x) T = N (discrete time) M = [0, 1] (state space)

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Examples

• 2. Finite dimensional continuous dynamical systems: ODEs

x0 Φ(t, x0) Φ(−t, x0)

Lorenz equations

⇥ f ∈ C1(Rn) ⇤

dx dt = f(x)

x(0) = x0

⇥ ⇧ ⇧ ⌅ ⇧ ⇧ ⇤

(IVP)

Φ(t, x0) : solution of the (IVP)

Φ : T × M → M

(t, x0) 7! Φ(t, x0)

(continuous time)

T = R

(state space)

M = Rn

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Examples

Φ : T × M → M

(infinite dimensional state space)

M = L2(Ω)

(continuous time)

T = [0, ∞)

(semigroup)

(a) Partial differential equations

• 3. Infinite dimensional continuous dynamical systems

Cahn-Hilliard equation

Ω ⊂ Rn, n = 1, 2, 3 ∂u ∂t − ∆

• −ν∆u − u + u3

= 0

(t, u0) 7! Φ(t, u0)

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Examples

(b) Delay differential equations y0(t) = F(y(t), y(t − τ))

Φ : T × M → M

(infinite dimensional state space) (continuous time)

T = [0, ∞)

(semigroup)

M = C[−τ, 0] (t, y0) 7! Φ(t, y0)

• 3. Infinite dimensional continuous dynamical systems

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Examples

(b) Delay differential equations y0(t) = F(y(t), y(t − τ))

Φ : T × M → M

(infinite dimensional state space) (continuous time)

T = [0, ∞)

(semigroup)

M = C[−τ, 0]

12 14 16 18 20 22 24 1 0.5 0.5 1 1.5 2 2.5 3 3.5

y0 Φ(t, y0)

ex: y′(t) = − 12

5 y(t − 1)[1 + y(t)]

(t, y0) 7! Φ(t, y0)

• 3. Infinite dimensional continuous dynamical systems

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What kind of solutions are we interested in ?

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In any dynamical system, it is the bounded solutions which are most important and which should be investigated first.

Henri Poincaré

What kind of solutions are we interested in ?

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In any dynamical system, it is the bounded solutions which are most important and which should be investigated first.

Henri Poincaré

Compact invariant sets

Exploit smoothness, boundedness and low dimensionality.

What kind of solutions are we interested in ?

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• Equilibrium solutions.
• Time periodic solutions.
• Connecting orbits.
• Global attractors.

In any dynamical system, it is the bounded solutions which are most important and which should be investigated first.

Henri Poincaré

Compact invariant sets

Exploit smoothness, boundedness and low dimensionality.

What kind of solutions are we interested in ?

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F(x) = 0

• Equilibrium solutions.
• Time periodic solutions.
• Connecting orbits.
• Global attractors.

In any dynamical system, it is the bounded solutions which are most important and which should be investigated first.

Henri Poincaré

Compact invariant sets

Exploit smoothness, boundedness and low dimensionality.

What kind of solutions are we interested in ?

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In practice, how to study a dynamical system?

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In practice, how to study a dynamical system?

A standard approach is to get insight from numerical simulations to formulate new conjectures, and then attempt to prove the conjectures using pure mathematical techniques only. Actually, this strong dichotomy need not exist in the context of dynamical systems, as the strength of numerical analysis and functional analysis can be combined to prove, in a rigorous mathematical sense, the existence of equilibria, periodic solutions, connecting orbits.... and even chaotic dynamics !

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In practice, how to study a dynamical system?

Rigorous computations

The goal of rigorous computations is to construct algorithms that provide an approximate solution to the problem together with precise and possibly efficient bounds within which the exact solution is guaranteed to exist in the mathematically rigorous sense. A standard approach is to get insight from numerical simulations to formulate new conjectures, and then attempt to prove the conjectures using pure mathematical techniques only. Actually, this strong dichotomy need not exist in the context of dynamical systems, as the strength of numerical analysis and functional analysis can be combined to prove, in a rigorous mathematical sense, the existence of equilibria, periodic solutions, connecting orbits.... and even chaotic dynamics !

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Motivation: how to study parameter dependent infinite dimensional problems ?

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• x1

x2 x3 x4 x5 x6 x7

X

F(x) = 0

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• x1

x2 x3 x4 x5 x6 x7

X

Often impossible to compute exactly !

F(x) = 0

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X

Alternative: find small balls in which it is demonstrated (in a mathematically rigorous sense) that a unique solution exists.

F(x) = 0

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Rigorous Computations

(Ingredients)

• 1. Smoothness of the solutions
• 2. Banach space of algebraically decaying sequences
• 3. Finite dimensional Galerkin projection
• 4. Bounds on the truncation error terms (Analytic estimates)
• 5. Fixed point theory, Uniform contraction principle
• 6. Numerical analysis (continuation, Fast Fourier transform)
• 7. Interval Arithmetic

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Rigorous Computations

(Ingredients)

• Predictor

Predictor Corrector

f(x, ν) = 0

||x||

ν

(x0, ν0) (x1, ν1)

(˜ x1, ν1)

˙ x

• Continuation

(Predictor-Corrector Algorithm)

• 1. Smoothness of the solutions
• 2. Banach space of algebraically decaying sequences
• 3. Finite dimensional Galerkin projection
• 4. Bounds on the truncation error terms (Analytic estimates)
• 5. Fixed point theory, Uniform contraction principle
• 6. Numerical analysis (continuation, Fast Fourier transform)
• 7. Interval Arithmetic

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Rigorous Computations

spectral method

⇥

f(x, ν) = 0

x ν

: modes : parameter

F(u, ν) = 0

(Differential Equation)

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Rigorous Computations

spectral method

⇥

f(x, ν) = 0

x ν

: modes : parameter

F(u, ν) = 0

(Differential Equation)

Knowledge about regularity ⇥ x 2 Ωs =

⇢ (xk)k : kxks = sup

k

{kxk∞ks} < 1

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Rigorous Computations

spectral method

⇥

f(x, ν) = 0

x ν

: modes : parameter

¯ x

Galerkin approximation

Consider such that . f (m)(¯ x, ν0) ≈ 0

Newton-like operator

¯ x

at

f(x, ν) = 0 ⇐ ⇒ Tν(x) = x

F(u, ν) = 0

(Differential Equation)

Knowledge about regularity ⇥ x 2 Ωs =

⇢ (xk)k : kxks = sup

k

{kxk∞ks} < 1

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Rigorous Computations

spectral method

⇥

f(x, ν) = 0

x ν

: modes : parameter

¯ x

Galerkin approximation

Consider such that . f (m)(¯ x, ν0) ≈ 0

Newton-like operator

¯ x

at

f(x, ν) = 0 ⇐ ⇒ Tν(x) = x

Tν(x) = x − Jf(x, ν) Tν : Ωs → Ωs J ≈ Dxf(¯ x, ν0)−1

The chances of contracting a small set B around depends on the magnitude of the eigenvalues of .

¯ x J

⇤ ⌃ ⌃ ⇧ ⌃ ⌃ ⌅

F(u, ν) = 0

(Differential Equation)

Knowledge about regularity ⇥ x 2 Ωs =

⇢ (xk)k : kxks = sup

k

{kxk∞ks} < 1

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¯ x

• Q: How to find a ball such that

B¯

x(r)

T : B¯

x(r) → B¯ x(r) is a contraction?

Ωs

• ˆ

x

B¯

x(r)

ν

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¯ x

• Q: How to find a ball such that

B¯

x(r)

T : B¯

x(r) → B¯ x(r) is a contraction?

Ωs

• ˆ

x

ν

B¯

x(r)

B¯

x(r) = ¯

x + B(r)

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Ball of radius r centered at 0 in the space Ωs

¯ x

• Q: How to find a ball such that

B¯

x(r)

T : B¯

x(r) → B¯ x(r) is a contraction?

Ωs

• ˆ

x

ν

B¯

x(r)

B¯

x(r) = ¯

x + B(r)

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Ball of radius r centered at 0 in the space Ωs

¯ x

• Q: How to find a ball such that

B¯

x(r)

T : B¯

x(r) → B¯ x(r) is a contraction?

Ωs

• ˆ

x

ν

B¯

x(r)

• Tν(¯

x) − ¯ x

• k
• +

sup

b,c∈B(r)

• DxTν(¯

x + b)c

• k
• − r

ωs

k

≤ pk(r)

Radii polynomials {pk(r)}k

A: : upper bounds satisfying

B¯

x(r) = ¯

x + B(r)

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Ball of radius r centered at 0 in the space Ωs

¯ x

• Q: How to find a ball such that

B¯

x(r)

T : B¯

x(r) → B¯ x(r) is a contraction?

Ωs

• ˆ

x

ν

B¯

x(r)

• Tν(¯

x) − ¯ x

• k
• +

sup

b,c∈B(r)

• DxTν(¯

x + b)c

• k
• − r

ωs

k

≤ pk(r)

Radii polynomials {pk(r)}k

A: : upper bounds satisfying

B¯

x(r) = ¯

x + B(r)

Lemma: If there exists such that for all , then there is a unique s.t. .

ˆ x ∈ B¯

x(r)

f(ˆ x, ν) = 0 r > 0 pk(r) < 0 k

• proof. Banach fixed point theorem.

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Suppose there exist A1, A2, . . . , An such that for every j ∈ {1, . . . , n} and every k ∈ Zd, we have that

• c(j)

k

• ≤ Aj

ωs

k

, Then, for any k ∈ Zd, we get that

• ⇥

c(1) ∗ · · · ∗ c(n)⇤

k

• ≤

⌅ ⌃

n j=1

Aj ⇧ ⌥ α(n)

k

ωs

k

.

Analytic estimates to construct the polynomials

• ⇥

c(1) ∗ · · · ∗ c(n)⇤

k

• =
• ⌦

k1+···+kn=k k1,...,kn∈Zd

c(1)

k1 · · · c(n) kn

• ≤

⌦

k1+···+kn=k k1,...,kn∈Zd

A1 ωs

k1

· · · An ωs

kn

= ⌅ ⌃

n

↵

j=1

Aj ⇧ ⌥ ⌅

• ⌃

⌦

k1+···+kn=k k1,...,kn∈Zd

1 ωs

k1 · · · ωs kn

⇧ ⌥ = ⌅ ⌃

n

↵

j=1

Aj ⇧ ⌥ ⌅

• ⌃

⌦

k1+···+kn=k k1,...,kn∈Zd d

↵

j=1

1 ωsj

k1 j · · · ωsj kn j

⇧ ⌥ = ⌅ ⌃

n

↵

j=1

Aj ⇧ ⌥ ⌅

• ⌃

d

↵

j=1

⌦

k1 j +···+kn j =kj k1 j ,...,kn j ∈Z

1 ωsj

k1 j · · · ωsj kn j

⇧ ⌥ ≤ ⌅ ⌃

n

↵

j=1

Aj ⇧ ⌥

d

↵

j=1

α(n)

kj

ωsj

kj

= ⌅ ⌃

n

↵

j=1

Aj ⇧ ⌥ α(n)

k

ωs

k

.

Proof.

• M. Gameiro & J.-P. L. Analytic estimates and rigorous continuation for equilibria of

higher-dimensional PDEs. Journal of Differential Equations, 2010.

ωs

k = |k1|s1 · · · |kd|sd

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The rigorous computational method

||x||s

• ¯

x •

• B¯

x(r)

xν = ¯ x + ∆ν ˙ x

f(x, ν) = 0

Bxν(r)

ν

ν0 ν0 + ∆ν

Radii polynomials {pk(r, ∆ν)}

Verifying the uniform contraction principle.

⇥

T ∃ r > 0 s.t. pk(r, ∆ν) < 0, ∀k = ⇒

: uniform contraction on [ν0, ν0 + ∆ν]

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Gluing the smooth pieces

¯ x0 ¯ x1

• ν0

ν1 ν2

• B1

B0

B+

1

B−

1

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Gluing the smooth pieces

¯ x0 ¯ x1

• ν0

ν1 ν2

• B1

B0

B+

1

B−

1

{(x, ν) | f(x, ν) = 0, ν ∈ [ν0, ν1]}

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¯ x0 ¯ x1

• ν0

ν1 ν2

• B1

B0

B+

1

B−

1

{(x, ν) | f(x, ν) = 0, ν ∈ [ν0, ν2]}

Gluing the smooth pieces

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¯ x0 ¯ x1

• ν0

ν1 ν2

• B1

B0

B+

1

B−

1

{(x, ν) | f(x, ν) = 0, ν ∈ [ν0, ν2]}

• Global smooth curves of solutions.
• Local uniqueness by the Banach fixed point theorem.
• Proof of non existence of secondary bifurcations along the curves.

Gluing the smooth pieces

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• Initial value problems of ODEs (Chebyshev in time)
• Boundary value problems of ODEs (Chebyshev in time)
• Periodic solutions of ODEs (Fourier in time)
• Connecting orbits of ODEs (Chebyshev in time + parameterization
• f invariant manifolds using power series)
• Equilibria of PDEs (Fourier in space)
• Periodic solutions of delay differential equations (Fourier in time)
• Minimizers of action functionals (Chebyshev in time)
• Periodic solutions of PDEs (Fourier in space and in time)

Applications

SLIDE 48

• Initial value problems of ODEs (Chebyshev in time)
• Boundary value problems of ODEs (Chebyshev in time)
• Periodic solutions of ODEs (Fourier in time)
• Connecting orbits of ODEs (Chebyshev in time + parameterization
• f invariant manifolds using power series)
• Equilibria of PDEs (Fourier in space)
• Periodic solutions of delay differential equations (Fourier in time)
• Minimizers of action functionals (Chebyshev in time)
• Periodic solutions of PDEs (Fourier in space and in time)

Applications

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• 1. Homoclinic and heteroclinic orbits
• f ODEs (traveling waves)

dx dt = f(x)

ODEs

lim

t→±∞ x(t) = x± ∈ Rn

x+ = x−

homoclinic orbit

x+ 6= x−

heteroclinic orbit

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Rigorous Computations

Connecting Orbits

Compute a set of equilibria.

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Rigorous Computations

Connecting Orbits

Compute a set of equilibria.

Parameterization method

Local representation of the invariant manifolds.

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Rigorous Computations

Connecting Orbits

Compute a set of equilibria.

Parameterization method

Local representation of the invariant manifolds. Connecting orbits between the equilibria?

Boundary value problem Chebyshev series Radii polynomials

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Ω = [0, π] × [0, π 1.001] × [0, π 1.002]

0.5 1 1.5 2 2.5 3 3.5 0.1 0.2 0.3 0.4 0.5 0.6

1

2

u

(1) (2) (3) (4) (5) (6) (7)

ν

Cahn-Hilliard 3D

• 2. Equilibria of PDEs

   ut = −∆( 1

ν ∆u + u − u3),

in Ω ∂u ∂n = ∂∆u ∂n = 0,

• n ∂Ω

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0.005 0.01 0.015 0.02 0.025 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

d

z(0)

⇤ ⌃ ⌃ ⌃ ⌃ ⌃ ⇧ ⌃ ⌃ ⌃ ⌃ ⌃ ⌅

∂tx = d∆x + (r1 − a1(x + y) − b1z)x + 1 ε

• y
• 1 − z

N

⇥

− x z N

⇥

, ∂ty = (d + βN)∆y + (r1 − a1(x + y) − b1z)y − 1 ε

• y
• 1 − z

N

⇥

− x z N

⇥

, ∂tz = d∆z + (r2 − b2(x + y) − a2z)z.

Systems of reaction-diffusion PDEs

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0.005 0.01 0.015 0.02 0.025 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

d

z(0)

⇤ ⌃ ⌃ ⌃ ⌃ ⌃ ⇧ ⌃ ⌃ ⌃ ⌃ ⌃ ⌅

∂tx = d∆x + (r1 − a1(x + y) − b1z)x + 1 ε

• y
• 1 − z

N

⇥

− x z N

⇥

, ∂ty = (d + βN)∆y + (r1 − a1(x + y) − b1z)y − 1 ε

• y
• 1 − z

N

⇥

− x z N

⇥

, ∂tz = d∆z + (r2 − b2(x + y) − a2z)z.

Systems of reaction-diffusion PDEs

• 11 co-existing steady

states at d = 0.006

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2.5 3 3.5 4 0.8 1 1.2 1.4 1.6 1.8 2

0.5 1 1.5 2 2.5 3 3.5 −1 1 2 3 4 5 6

kxk`2

0.5 1 1.5 2 2.5 3 3.5 −1 −0.5 0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 3.5 −1 1 2 3 4 5 6 7

x1 x2 x3

f(x, ν) = 0 ν

• 3. Periodic solutions of delay equations

y0(t) = F (y(t), y(t − τ1), . . . , y(t − τd)) ,

y0(t) = −ν [y(t − τ1) + y(t − τ2)] [1 + y(t)],

SLIDE 57

0.05 0.1 0.15 0.2 0.25 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

kxk`2

0.5 1 1.5 2 2.5 3 3.5 4 −1 −0.5 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 3 3.5 4 −1 −0.5 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 3.5 4 −1 1 2 3 4 5

y0(t) = − [2.425y(t − τ1) + 2.425y(t − τ2) + νy(t − τ3)] [1 + y(t)],

ν

SLIDE 58

Ginzburg–Landau energy: a model of superconductivity

G = G(φ, a) = 1 2d Z d

d

• φ2(φ2 − 2) + 2(φ0)2

κ2 + 2φ2a2 + 2(a0 − he)2 dt.

• φ > 0

a

• 4. Minimizers of action functionals

SLIDE 59

Ginzburg–Landau energy: a model of superconductivity

G = G(φ, a) = 1 2d Z d

d

• φ2(φ2 − 2) + 2(φ0)2

κ2 + 2φ2a2 + 2(a0 − he)2 dt.

• φ > 0

a

• Parameters
• d

he κ

• 4. Minimizers of action functionals

SLIDE 60

Ginzburg–Landau energy: a model of superconductivity

G = G(φ, a) = 1 2d Z d

d

• φ2(φ2 − 2) + 2(φ0)2

κ2 + 2φ2a2 + 2(a0 − he)2 dt.

• φ > 0

a

• Parameters
• d

he κ

• 4. Minimizers of action functionals

he φ(d)

0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 h Bifurcation diagram for kappa = 0.3, d = 4 Bifurcation Asym Sym

κ = 0.3, d = 4

Co-existence of nontrivial solutions

SLIDE 61

Kuramoto-Sivashinski equation (KS)

( ut = −νuyyyy − uyy + 2uuy u(t, y) = u(t, y + 2π), u(t, −y) = −u(t, y)

• 5. Periodic orbits of PDEs

SLIDE 62

Kuramoto-Sivashinski equation

Popular model to analyze weak turbulence

• r spatiotemporal chaos

(KS)

( ut = −νuyyyy − uyy + 2uuy u(t, y) = u(t, y + 2π), u(t, −y) = −u(t, y)

• 5. Periodic orbits of PDEs

SLIDE 63

Kuramoto-Sivashinski equation

Popular model to analyze weak turbulence

• r spatiotemporal chaos

A common approach to study time-periodic solutions of (KS) is to construct a Poincaré map via numerical integration of the flow, and to look for fixed points of this map on a prescribed Poincaré section.

(KS)

( ut = −νuyyyy − uyy + 2uuy u(t, y) = u(t, y + 2π), u(t, −y) = −u(t, y)

• 5. Periodic orbits of PDEs

SLIDE 64

Kuramoto-Sivashinski equation

Popular model to analyze weak turbulence

• r spatiotemporal chaos

Christiansen, Cvitanovic, Lan, Johnson, Jolly, Kevrekidis, Putkaradze, ...

A common approach to study time-periodic solutions of (KS) is to construct a Poincaré map via numerical integration of the flow, and to look for fixed points of this map on a prescribed Poincaré section.

(KS)

( ut = −νuyyyy − uyy + 2uuy u(t, y) = u(t, y + 2π), u(t, −y) = −u(t, y)

• 5. Periodic orbits of PDEs

SLIDE 65

Kuramoto-Sivashinski equation

Popular model to analyze weak turbulence

• r spatiotemporal chaos

Christiansen, Cvitanovic, Lan, Johnson, Jolly, Kevrekidis, Putkaradze, ...

A common approach to study time-periodic solutions of (KS) is to construct a Poincaré map via numerical integration of the flow, and to look for fixed points of this map on a prescribed Poincaré section.

Goal: propose an method (based on spectral methods and fixed point theory) to rigorously compute time periodic solutions of PDEs.

(KS)

( ut = −νuyyyy − uyy + 2uuy u(t, y) = u(t, y + 2π), u(t, −y) = −u(t, y)

• 5. Periodic orbits of PDEs

SLIDE 66

Letting L = 2π

p , the time-periodic solutions of period p of (KS) can be expanded

using the Fourier expansion u(t, y) = X

k∈Z2

ckψk, where for k = (k1, k2) ∈ Z2, ψk = eiLk1teik2y.

SLIDE 67

Letting L = 2π

p , the time-periodic solutions of period p of (KS) can be expanded

using the Fourier expansion u(t, y) = X

k∈Z2

ckψk, where for k = (k1, k2) ∈ Z2, ψk = eiLk1teik2y.

ak

def

= Re(ck) and bk

def

= Im(ck). xk = 8 > > < > > : L, k = (0, 0) bk, k = (0, k2), k2 6= 0 ✓

ak bk

◆ , k = (k1, k2), k1 6= 0 and k2 6= 0.

SLIDE 68

Letting L = 2π

p , the time-periodic solutions of period p of (KS) can be expanded

using the Fourier expansion u(t, y) = X

k∈Z2

ckψk, where for k = (k1, k2) ∈ Z2, ψk = eiLk1teik2y.

ak

def

= Re(ck) and bk

def

= Im(ck). xk = 8 > > < > > : L, k = (0, 0) bk, k = (0, k2), k2 6= 0 ✓

ak bk

◆ , k = (k1, k2), k1 6= 0 and k2 6= 0. Unknowns

SLIDE 69

Letting L = 2π

p , the time-periodic solutions of period p of (KS) can be expanded

using the Fourier expansion u(t, y) = X

k∈Z2

ckψk, where for k = (k1, k2) ∈ Z2, ψk = eiLk1teik2y.

ak

def

= Re(ck) and bk

def

= Im(ck). xk = 8 > > < > > : L, k = (0, 0) bk, k = (0, k2), k2 6= 0 ✓

ak bk

◆ , k = (k1, k2), k1 6= 0 and k2 6= 0. Unknowns

F

Plugging the space-time Fourier expansion into (KS) results in solving, for all k 2 Z2 hk

def

= µkck 2 X

k1+k2=k

ik1

2ck1ck2 = µkck k2i

X

k1+k2=k

ck1ck2 = 0, where µk = µk1,k2

def

= ik1L + νk4

2 k2 2.

SLIDE 70

Letting L = 2π

p , the time-periodic solutions of period p of (KS) can be expanded

using the Fourier expansion u(t, y) = X

k∈Z2

ckψk, where for k = (k1, k2) ∈ Z2, ψk = eiLk1teik2y.

ak

def

= Re(ck) and bk

def

= Im(ck). xk = 8 > > < > > : L, k = (0, 0) bk, k = (0, k2), k2 6= 0 ✓

ak bk

◆ , k = (k1, k2), k1 6= 0 and k2 6= 0. fk

def

= Re(hk) =

• νk4

2 k2 2

• ak (k1L)bk + 2k2

X

k1+k2=k

ak1bk2, gk

def

= Im(hk) = (k1L)ak +

• νk4

2 k2 2

• bk k2

X

k1+k2=k

(ak1ak2 bk1bk2). Unknowns

F

Plugging the space-time Fourier expansion into (KS) results in solving, for all k 2 Z2 hk

def

= µkck 2 X

k1+k2=k

ik1

2ck1ck2 = µkck k2i

X

k1+k2=k

ck1ck2 = 0, where µk = µk1,k2

def

= ik1L + νk4

2 k2 2.

SLIDE 71

Letting L = 2π

p , the time-periodic solutions of period p of (KS) can be expanded

using the Fourier expansion u(t, y) = X

k∈Z2

ckψk, where for k = (k1, k2) ∈ Z2, ψk = eiLk1teik2y.

ak

def

= Re(ck) and bk

def

= Im(ck). xk = 8 > > < > > : L, k = (0, 0) bk, k = (0, k2), k2 6= 0 ✓

ak bk

◆ , k = (k1, k2), k1 6= 0 and k2 6= 0. fk

def

= Re(hk) =

• νk4

2 k2 2

• ak (k1L)bk + 2k2

X

k1+k2=k

ak1bk2, gk

def

= Im(hk) = (k1L)ak +

• νk4

2 k2 2

• bk k2

X

k1+k2=k

(ak1ak2 bk1bk2). Unknowns Functions

F

Plugging the space-time Fourier expansion into (KS) results in solving, for all k 2 Z2 hk

def

= µkck 2 X

k1+k2=k

ik1

2ck1ck2 = µkck k2i

X

k1+k2=k

ck1ck2 = 0, where µk = µk1,k2

def

= ik1L + νk4

2 k2 2.

SLIDE 72

Defining I = {(0, 0)} [ {k = (0, k2) | k2 6= 0} [ {k = (k1, k2) | k1 6= 0 and k2 6= 0},

• ne can identify x = {xk}k∈I.

xk = 8 > > < > > : L, k = (0, 0) bk, k = (0, k2), k2 6= 0 ✓

ak bk

◆ , k = (k1, k2), k1 6= 0 and k2 6= 0. Unknowns

SLIDE 73

• X

Finally, let us define F = {Fk}k∈I component-wise by Fk = 8 > > < > > : η, k = (0, 0) gk, k = (0, k2), k2 6= 0 ✓

fk gk

◆ , k = (k1, k2), k1 6= 0 and k2 6= 0.

Defining I = {(0, 0)} [ {k = (0, k2) | k2 6= 0} [ {k = (k1, k2) | k1 6= 0 and k2 6= 0},

• ne can identify x = {xk}k∈I.

xk = 8 > > < > > : L, k = (0, 0) bk, k = (0, k2), k2 6= 0 ✓

ak bk

◆ , k = (k1, k2), k1 6= 0 and k2 6= 0. Unknowns

SLIDE 74

• X

Finally, let us define F = {Fk}k∈I component-wise by Fk = 8 > > < > > : η, k = (0, 0) gk, k = (0, k2), k2 6= 0 ✓

fk gk

◆ , k = (k1, k2), k1 6= 0 and k2 6= 0. : Hence,

• Lemma. Finding time-periodic solutions u(t, y) of (KS) such that η = 0

is equivalent to find x such that F(x) = 0.

Defining I = {(0, 0)} [ {k = (0, k2) | k2 6= 0} [ {k = (k1, k2) | k1 6= 0 and k2 6= 0},

• ne can identify x = {xk}k∈I.

xk = 8 > > < > > : L, k = (0, 0) bk, k = (0, k2), k2 6= 0 ✓

ak bk

◆ , k = (k1, k2), k1 6= 0 and k2 6= 0. Unknowns

SLIDE 75

To solve rigorously in a Banach space

• X

Finally, let us define F = {Fk}k∈I component-wise by Fk = 8 > > < > > : η, k = (0, 0) gk, k = (0, k2), k2 6= 0 ✓

fk gk

◆ , k = (k1, k2), k1 6= 0 and k2 6= 0. : Hence,

• Lemma. Finding time-periodic solutions u(t, y) of (KS) such that η = 0

is equivalent to find x such that F(x) = 0.

Defining I = {(0, 0)} [ {k = (0, k2) | k2 6= 0} [ {k = (k1, k2) | k1 6= 0 and k2 6= 0},

• ne can identify x = {xk}k∈I.

xk = 8 > > < > > : L, k = (0, 0) bk, k = (0, k2), k2 6= 0 ✓

ak bk

◆ , k = (k1, k2), k1 6= 0 and k2 6= 0. Unknowns

SLIDE 76

j

  Define the one-dimensional weights ωs

k by

ωs

k

def

= ( 1, if k = 0 |k|s, if k 6= 0. Using the 1-d weights, define the 2-dimensional weights, given k = (k1, k2) 2 Z2, ωs

k

def

= ωs1

k1ωs2 k2.

They are used to define the norm kxks = sup

k∈I

ωs

k|xk|∞,

where |xk|∞ is the sup norm of the vector xk, which is one or two dimensional, depending

• n k. Define the Banach space

Xs = {x | kxks < 1} , consisting of sequences with algebraically decaying tails according to the rate s.

The Banach space

SLIDE 77

Banach algebra under discrete convolution

j

  Define the one-dimensional weights ωs

k by

ωs

k

def

= ( 1, if k = 0 |k|s, if k 6= 0. Using the 1-d weights, define the 2-dimensional weights, given k = (k1, k2) 2 Z2, ωs

k

def

= ωs1

k1ωs2 k2.

They are used to define the norm kxks = sup

k∈I

ωs

k|xk|∞,

where |xk|∞ is the sup norm of the vector xk, which is one or two dimensional, depending

• n k. Define the Banach space

Xs = {x | kxks < 1} , consisting of sequences with algebraically decaying tails according to the rate s.

The Banach space

SLIDE 78

For sake of simplicity of the presentation, for k = (k1, k2) with k1 6= 0 or k2 6= 0, let Rk(ν, L)

def

= ✓ νk4

2 k2 2

k1L k1L νk4

2 k2 2

◆ and R0,k2(ν, L)

def

= νk4

2 k2 2,

Nk(x)

def

= X

k1+k2=k

✓ 2ak1bk2 ak1ak2 + bk1bk2 ◆ so that one has that Fk(x, ν) = Rk(ν, L)xk + k2Nk(x). Z

SLIDE 79

For sake of simplicity of the presentation, for k = (k1, k2) with k1 6= 0 or k2 6= 0, let Rk(ν, L)

def

= ✓ νk4

2 k2 2

k1L k1L νk4

2 k2 2

◆ and R0,k2(ν, L)

def

= νk4

2 k2 2,

Nk(x)

def

= X

k1+k2=k

✓ 2ak1bk2 ak1ak2 + bk1bk2 ◆ so that one has that Fk(x, ν) = Rk(ν, L)xk + k2Nk(x). Z

SLIDE 80

For sake of simplicity of the presentation, for k = (k1, k2) with k1 6= 0 or k2 6= 0, let Rk(ν, L)

def

= ✓ νk4

2 k2 2

k1L k1L νk4

2 k2 2

◆ and R0,k2(ν, L)

def

= νk4

2 k2 2,

Nk(x)

def

= X

k1+k2=k

✓ 2ak1bk2 ak1ak2 + bk1bk2 ◆ so that one has that Fk(x, ν) = Rk(ν, L)xk + k2Nk(x). Z

• Lemma. (Bootstrap) Consider a fixed decay rate s > (1, 1) and assume the existence of

M > (0, 0) such that Rk(ν, L) is invertible for all |k| > M. If there exists x ∈ Xs such that F(x) = 0, then x ∈ Xs0, for all s0 > (1, 1).

SLIDE 81

For sake of simplicity of the presentation, for k = (k1, k2) with k1 6= 0 or k2 6= 0, let Rk(ν, L)

def

= ✓ νk4

2 k2 2

k1L k1L νk4

2 k2 2

◆ and R0,k2(ν, L)

def

= νk4

2 k2 2,

Nk(x)

def

= X

k1+k2=k

✓ 2ak1bk2 ak1ak2 + bk1bk2 ◆ so that one has that Fk(x, ν) = Rk(ν, L)xk + k2Nk(x). Z

• Lemma. (Bootstrap) Consider a fixed decay rate s > (1, 1) and assume the existence of

M > (0, 0) such that Rk(ν, L) is invertible for all |k| > M. If there exists x ∈ Xs such that F(x) = 0, then x ∈ Xs0, for all s0 > (1, 1).

Hence, we focus our attention on looking for zeros of F within a Banach space with a fixed decay rate s>(1,1).

SLIDE 82

Given m = (m1, m2), define Fm = Fm1 ⇥ Fm2, where Fmj

def

= {kj 2 Z | |kj| < mj}. Consider a Galerkin projection of F of dimension n = n(m)

def

= 2m1m2 2m1 m2 + 2 given by F(m)

def

= {F(m)

k

}k∈Fm, where F(m) : Rn ! Rn, is given component-wise by F(m)

k

(xFm)

def

= Fk(xFm, 0Im), k 2 Fm.

SLIDE 83

Given m = (m1, m2), define Fm = Fm1 ⇥ Fm2, where Fmj

def

= {kj 2 Z | |kj| < mj}. Consider a Galerkin projection of F of dimension n = n(m)

def

= 2m1m2 2m1 m2 + 2 given by F(m)

def

= {F(m)

k

}k∈Fm, where F(m) : Rn ! Rn, is given component-wise by F(m)

k

(xFm)

def

= Fk(xFm, 0Im), k 2 Fm. Consider ˆ xFm such that F(m)(ˆ xFm) ⇡ 0. Let ˆ x

def

= (ˆ xFm, 0Im) 2 Xs. Assume that the Jacobian matrix DF(m)(ˆ xFm) is non-singular and let Am an approximation for its inverse.

SLIDE 84

Given m = (m1, m2), define Fm = Fm1 ⇥ Fm2, where Fmj

def

= {kj 2 Z | |kj| < mj}. Consider a Galerkin projection of F of dimension n = n(m)

def

= 2m1m2 2m1 m2 + 2 given by F(m)

def

= {F(m)

k

}k∈Fm, where F(m) : Rn ! Rn, is given component-wise by F(m)

k

(xFm)

def

= Fk(xFm, 0Im), k 2 Fm. Consider ˆ xFm such that F(m)(ˆ xFm) ⇡ 0. Let ˆ x

def

= (ˆ xFm, 0Im) 2 Xs. Assume that the Jacobian matrix DF(m)(ˆ xFm) is non-singular and let Am an approximation for its inverse. Define the action of the linear operator A on x = {xk}k∈I component-wise by h A(x) i

k

def

= 8 < : h Am(xFm) i

k,

if k 2 Fm Rk(ν, ˆ L)−1xk, if k 62 Fm. T(x)

def

= x AF(x).

SLIDE 85

(Newton-like operator)

Given m = (m1, m2), define Fm = Fm1 ⇥ Fm2, where Fmj

def

= {kj 2 Z | |kj| < mj}. Consider a Galerkin projection of F of dimension n = n(m)

def

= 2m1m2 2m1 m2 + 2 given by F(m)

def

= {F(m)

k

}k∈Fm, where F(m) : Rn ! Rn, is given component-wise by F(m)

k

(xFm)

def

= Fk(xFm, 0Im), k 2 Fm. Consider ˆ xFm such that F(m)(ˆ xFm) ⇡ 0. Let ˆ x

def

= (ˆ xFm, 0Im) 2 Xs. Assume that the Jacobian matrix DF(m)(ˆ xFm) is non-singular and let Am an approximation for its inverse. Define the action of the linear operator A on x = {xk}k∈I component-wise by h A(x) i

k

def

= 8 < : h Am(xFm) i

k,

if k 2 Fm Rk(ν, ˆ L)−1xk, if k 62 Fm. T(x)

def

= x AF(x).

SLIDE 86

(Newton-like operator)

Given m = (m1, m2), define Fm = Fm1 ⇥ Fm2, where Fmj

def

= {kj 2 Z | |kj| < mj}. Consider a Galerkin projection of F of dimension n = n(m)

def

= 2m1m2 2m1 m2 + 2 given by F(m)

def

= {F(m)

k

}k∈Fm, where F(m) : Rn ! Rn, is given component-wise by F(m)

k

(xFm)

def

= Fk(xFm, 0Im), k 2 Fm.

• F
• Lemma. Consider a Galerkin projection dimension m = (m1, m2) and let s = (s1, s2) >

(1, 1) a decay rate. The solutions of F = 0 are in one to one correspondence with the fixed points of T. Also, one has that T : Xs ! Xs. Consider ˆ xFm such that F(m)(ˆ xFm) ⇡ 0. Let ˆ x

def

= (ˆ xFm, 0Im) 2 Xs. Assume that the Jacobian matrix DF(m)(ˆ xFm) is non-singular and let Am an approximation for its inverse. Define the action of the linear operator A on x = {xk}k∈I component-wise by h A(x) i

k

def

= 8 < : h Am(xFm) i

k,

if k 2 Fm Rk(ν, ˆ L)−1xk, if k 62 Fm. T(x)

def

= x AF(x).

SLIDE 87

The rigorous continuation method is based on the notion of the radii polynomials, which provide a numerically efficient way to verify that the operator T is a contraction on a small closed ball B(ˆ x, r) centered at the numerical approximation ˆ x in Xs.

SLIDE 88

Ingredients to construct the radii polynomials

• Convolution estimates
• Interval arithmetic
• Fast Fourier transform

The rigorous continuation method is based on the notion of the radii polynomials, which provide a numerically efficient way to verify that the operator T is a contraction on a small closed ball B(ˆ x, r) centered at the numerical approximation ˆ x in Xs.

SLIDE 89

Ingredients to construct the radii polynomials

• Convolution estimates
• Interval arithmetic
• Fast Fourier transform

The rigorous continuation method is based on the notion of the radii polynomials, which provide a numerically efficient way to verify that the operator T is a contraction on a small closed ball B(ˆ x, r) centered at the numerical approximation ˆ x in Xs. The closed ball of radius r in Xs, centered at the origin, is given by B(r)

def

= Y

k∈I

 r ωs

k

, r ωs

k

d(k) , where d(k) = 1 if k = (0, k2) and d(k) = 2 otherwise. The closed ball of radius r centered at ˆ x is then B(ˆ x, r)

def

= ˆ x + B(r).

SLIDE 90

Consider now bounds Yk and Zk for all k 2 I, such that

• ⇥

T(ˆ x) ˆ x ⇤

k

•  Yk,

and sup

x1,x2∈B(r)

• ⇥

DT(ˆ x + x1)x2 ⇤

k

•  Zk(r).

Lemma. If there exists an r > 0 such that kY + Zks < r, with Y

def

= {Yk}k∈I and Z

def

= {Zk}k∈I, then T is a contraction mapping on B(ˆ x, r) with contraction constant at most kY + Zks/r < 1. Furthermore, there is a unique ˜ x 2 B(ˆ x, r) such that F(˜ x) = 0.

SLIDE 91

Define the finite radii polynomials {pk(r)}k∈FM by pk(r)

def

= Yk + Zk(r) r !s

k

Id(k), and the tail radii polynomial by ˜ pM(r)

def

= ˜ ZM(r) 1. Consider now bounds Yk and Zk for all k 2 I, such that

• ⇥

T(ˆ x) ˆ x ⇤

k

•  Yk,

and sup

x1,x2∈B(r)

• ⇥

DT(ˆ x + x1)x2 ⇤

k

•  Zk(r).

Lemma. If there exists an r > 0 such that kY + Zks < r, with Y

def

= {Yk}k∈I and Z

def

= {Zk}k∈I, then T is a contraction mapping on B(ˆ x, r) with contraction constant at most kY + Zks/r < 1. Furthermore, there is a unique ˜ x 2 B(ˆ x, r) such that F(˜ x) = 0.

SLIDE 92

Define the finite radii polynomials {pk(r)}k∈FM by pk(r)

def

= Yk + Zk(r) r !s

k

Id(k), and the tail radii polynomial by ˜ pM(r)

def

= ˜ ZM(r) 1. Consider now bounds Yk and Zk for all k 2 I, such that

• ⇥

T(ˆ x) ˆ x ⇤

k

•  Yk,

and sup

x1,x2∈B(r)

• ⇥

DT(ˆ x + x1)x2 ⇤

k

•  Zk(r).

Lemma. If there exists an r > 0 such that kY + Zks < r, with Y

def

= {Yk}k∈I and Z

def

= {Zk}k∈I, then T is a contraction mapping on B(ˆ x, r) with contraction constant at most kY + Zks/r < 1. Furthermore, there is a unique ˜ x 2 B(ˆ x, r) such that F(˜ x) = 0.

asymptotic bound for Z in X

k s

SLIDE 93

Define the finite radii polynomials {pk(r)}k∈FM by pk(r)

def

= Yk + Zk(r) r !s

k

Id(k), and the tail radii polynomial by ˜ pM(r)

def

= ˜ ZM(r) 1.

• Lemma. If there exists r > 0 such that pk(r) < 0 for all k 2 FM and ˜

pM(r) < 0, then there is a unique ˜ x 2 B(ˆ x, r) such that F(˜ x) = 0. Consider now bounds Yk and Zk for all k 2 I, such that

• ⇥

T(ˆ x) ˆ x ⇤

k

•  Yk,

and sup

x1,x2∈B(r)

• ⇥

DT(ˆ x + x1)x2 ⇤

k

•  Zk(r).

Lemma. If there exists an r > 0 such that kY + Zks < r, with Y

def

= {Yk}k∈I and Z

def

= {Zk}k∈I, then T is a contraction mapping on B(ˆ x, r) with contraction constant at most kY + Zks/r < 1. Furthermore, there is a unique ˜ x 2 B(ˆ x, r) such that F(˜ x) = 0.

asymptotic bound for Z in X

k s

SLIDE 94

Results

Kuramoto-Sivashinski equation (KS)

( ut = −νuyyyy − uyy + 2uuy u(t, y) = u(t, y + 2π), u(t, −y) = −u(t, y)

SLIDE 95

Results

Kuramoto-Sivashinski equation (KS)

( ut = −νuyyyy − uyy + 2uuy u(t, y) = u(t, y + 2π), u(t, −y) = −u(t, y)

m = (77, 15), M = (229, 43), s = ( 3

2, 3 2)

# of time Fourier modes # of space Fourier modes decay rates

SLIDE 96

Results

Kuramoto-Sivashinski equation (KS)

( ut = −νuyyyy − uyy + 2uuy u(t, y) = u(t, y + 2π), u(t, −y) = −u(t, y)

m = (77, 15), M = (229, 43), s = ( 3

2, 3 2)

# of time Fourier modes # of space Fourier modes decay rates

ν ∈ {.127, .12707, .12715, .12725, .12739, .12756, .12777}

SLIDE 97

Results

Kuramoto-Sivashinski equation (KS)

( ut = −νuyyyy − uyy + 2uuy u(t, y) = u(t, y + 2π), u(t, −y) = −u(t, y)

m = (77, 15), M = (229, 43), s = ( 3

2, 3 2)

# of time Fourier modes # of space Fourier modes decay rates

ν ∈ {.127, .12707, .12715, .12725, .12739, .12756, .12777}

Y

∈I

˜ x ∈ B(ˆ x, r) = ˆ x + Y

k∈I

" −3 × 10−4 k3/2

1

k3/2

2

, 3 × 10−4 k3/2

1

k3/2

2

#d(k) ⊂ X( 3

2 , 3 2 )

SLIDE 98

SLIDE 99

SLIDE 100

SLIDE 101

SLIDE 102

SLIDE 103

SLIDE 104

SLIDE 105

Thanks to my collaborators

• Maxime Breden (ENS, France)
• Roberto Castelli (BCAM, Spain)
• Anaïs Correc (Laval, Canada)
• Sarah Day (William & Mary, USA)
• Marcio Gameiro (Sao Paulo, Brazil)
• Gabor Kiss (Durham, UK)
• Konstantin Mischaikow (Rutgers, USA)
• Jason Mireles James (Rutgers, USA)
• Alessandro Pugliese (Bari, Italy)
• Christian Reinhardt (TU Munich, Germany)
• Matthieu

Vanicat (ENS, France)